Randomized Controlling Arrivals for a Finite Queue with Startup

In this work, we consider a randomized server control problem of an M/M/1/K queue with a removable server. When the number of customers reaches its capacity K ( K <; ∞), no further arriving customers are allowed to enter the system. Customers are allowed to the system with probability p and the customers are still unable to enter the system with probability 1 - p as the queue length decreases to a certain threshold value F. The server needs a startup time before allowing customers to enter the system. For this system, we develop the steady-state probability distribution of the number of customers in the system. Finally, we obtain the system characteristics. Some numerical results are also provided to demonstrate the results.